Sign test
Encyclopedia
In statistics
, the sign test can be used to test the hypothesis that there is "no difference in medians" between the continuous distributions of two random variable
s X and Y, in the situation when we can draw paired samples
from X and Y. It is a non-parametric test which makes very few assumptions about the nature of the distributions under test - this means that it has very general applicability but may lack the statistical power
of other tests such as the paired-samples t-test.
H0: p = 0.50. In other words, the null hypothesis states that given a random pair of measurements (xi, yi), then xi and yi are equally likely to be larger than the other.
To test the null hypothesis, independent pairs of sample data are collected from the populations {(x1, y1), (x2, y2), . . ., (xn, yn)}. Pairs are omitted for which there is no difference so that there is a possibility of a reduced sample of m pairs.
Then let w be the number of pairs for which yi − xi > 0. Assuming that H0 is true, then W follows a binomial distribution W ~ b(m, 0.5). The "W" is for Frank Wilcoxon
who developed the test, then later, the more powerful Wilcoxon signed-rank test
.
is used to calculate significance. The normal approximation to the binomial distribution can be used for large sample sizes, m>25.
The left-tail value is computed by Pr(W ≤ w), which is the p-value
for the alternative H1: p < 0.50. This alternative means that the X measurements tend to be higher.
The right-tail value is computed by Pr(W ≥ w), which is the p-value for the alternative H1: p > 0.50. This alternative means that the Y measurements tend to be higher.
For a two-sided alternative H1 the p-value is twice the smaller tail-value.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the sign test can be used to test the hypothesis that there is "no difference in medians" between the continuous distributions of two random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s X and Y, in the situation when we can draw paired samples
Paired difference test
In statistics, a paired difference test is a type of location test that is used when comparing two sets of measurements to assess whether their population means differ...
from X and Y. It is a non-parametric test which makes very few assumptions about the nature of the distributions under test - this means that it has very general applicability but may lack the statistical power
Statistical power
The power of a statistical test is the probability that the test will reject the null hypothesis when the null hypothesis is actually false . The power is in general a function of the possible distributions, often determined by a parameter, under the alternative hypothesis...
of other tests such as the paired-samples t-test.
Method
Let p = Pr(X > Y), and then test the null hypothesisNull hypothesis
The practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...
H0: p = 0.50. In other words, the null hypothesis states that given a random pair of measurements (xi, yi), then xi and yi are equally likely to be larger than the other.
To test the null hypothesis, independent pairs of sample data are collected from the populations {(x1, y1), (x2, y2), . . ., (xn, yn)}. Pairs are omitted for which there is no difference so that there is a possibility of a reduced sample of m pairs.
Then let w be the number of pairs for which yi − xi > 0. Assuming that H0 is true, then W follows a binomial distribution W ~ b(m, 0.5). The "W" is for Frank Wilcoxon
Frank Wilcoxon
Frank Wilcoxon was a chemist and statistician, known for the development of several statistical tests....
who developed the test, then later, the more powerful Wilcoxon signed-rank test
Wilcoxon signed-rank test
The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used when comparing two related samples or repeated measurements on a single sample to assess whether their population mean ranks differ The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used...
.
Assumptions
Let Zi = Yi – Xi for i = 1, ... , n.- The differences Zi are assumed to be independent.
- Each Zi comes from the same continuous population.
- The values of Xi and Yi represent are ordered (at least the ordinal scale), so the comparisons "greater than", "less than", and "equal to" are meaningful.
Significance testing
Since the test statistic is expected to follow a binomial distribution, the standard binomial testBinomial test
In statistics, the binomial test is an exact test of the statistical significance of deviations from a theoretically expected distribution of observations into two categories.-Common use:...
is used to calculate significance. The normal approximation to the binomial distribution can be used for large sample sizes, m>25.
The left-tail value is computed by Pr(W ≤ w), which is the p-value
P-value
In statistical significance testing, the p-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. One often "rejects the null hypothesis" when the p-value is less than the significance level α ,...
for the alternative H1: p < 0.50. This alternative means that the X measurements tend to be higher.
The right-tail value is computed by Pr(W ≥ w), which is the p-value for the alternative H1: p > 0.50. This alternative means that the Y measurements tend to be higher.
For a two-sided alternative H1 the p-value is twice the smaller tail-value.
See also
- Wilcoxon signed-rank testWilcoxon signed-rank testThe Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used when comparing two related samples or repeated measurements on a single sample to assess whether their population mean ranks differ The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used...
- A more powerful variant of the sign test, but one which also assumes a symmetric distribution. - Median testMedian testIn statistics, Mood's median test is a special case of Pearson's chi-squared test. It is a nonparametric test that tests the null hypothesis that the medians of the populations from which two samples are drawn are identical...
- An unpaired alternative to the sign test.